An easy way to quasiperiodic tilings

We want to make a tiling with 8-fold rotational symmetry. Thus we draw two square grids, one rotated by 45 degrees with respect to the other:

Two square grids, the small blue dots show the centers of the squares.

If you turn this drawing by 45 degrees you get essentially the same image again. Thus it has 8-fold rotational symmetry but it is clearly not a tiling. Looking closer we see small distinct parts of the drawing, which are repeated throughout. Particularly, there are 8-pointed stars formed by two overlapping squares. These stars need not to be exact, instead the centers of the two squares must be inside the other square and the corners of the squares have to be outside of the other square. Each of these stars gives us a point and thus we get this :

The centers of 8-pointed stars are shown by the large green dots.

Note that the large green dots form a rather regular and somehow repeating pattern. Always the same distances appear between them. Same dots correspond to the corners of squares. Other dots form a rhomb with the same sides as the squares and an acute angle of 45 degrees. We connect the dots with straight lines to make the rhombs and squares appear:

Lines between the green dots show the squares and rhombs.

We now fill the rhombs and squares with solid colors. Thus we get the quasiperiodic Amman-Beenker Tiling :

The final tiling. At the top and left the original square grids are left uncovered.

This method appears to be rather simple. But I do not use it just with ruler, compasses and the eye to draw manually such a tiling. It simply is too difficult to see if two squares form a star. Instead I use the computer.

In the next post I will discuss the mathematical details. You can then easily make a computer program for this method. We will see that this method is equal to the projection method.

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